Magnetic Field of a Circular Coil Lab 12

Size: px
Start display at page:

Download "Magnetic Field of a Circular Coil Lab 12"

Transcription

1 HB Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment- coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator, 3 leads Comment The computer monitor should be off to a void picking up an interference signal with the search coil. Reading Electrical Safety at the beginning of this manual. 1 Introduction A wire coil that is carrying a current produces a magnetic field B( r), where r is the distance from the center of the coil to the field point. The strength of the field B is proportional to the current I in the coil. The strength and direction of the field depend on r. For large distances from the coil (r a, wherea is the radius of the coil), the shape of the magnetic field of a coil is identical to the electric field produced by a point electric dipole. For large distances both fields fall off as 1/r 3. In this experiment you will measure the magnetic field of a circular coil at distances that are fairly close to the coil. The large distance approximation is not valid. A constant magnetic field can be measured in many ways. You can use a compass, a Hall Probe, a rotating coil of wire, or nuclear magnetic resonance. In this experiment the magnetic field will not be constant but will vary sinusoidally with time. Such a time varying magnetic field will induce a time varying voltage in a small coil which will be called the search coil. The search coil will be used to measure the magnetic field produced by a larger coil called the field coil. The current in the field coil will be varied sinusoidally with time and produce a sinusoidally varying magnetic field. 2 Electromagnetic Induction A magnetic field can be described at each point in space and time by a vector B whose direction coincides with the direction of the field and whose length is proportional to the magnitude B of the field. A magnetic field that changes with time produces a non-conservative electric field. This phenomenon, called electromagnetic induction, was discovered by Faraday, Henry, and others. The non-conservative electric field will produce a current and voltage in the search coil. By measuring this voltage for different positions and orientations of the small search coil the time varying magnetic field produced by the large field coil can be mapped out. The frequency of the sinusoidal current will be low enough so that the magnetic field mapped out by the search coil will be essentially identical to the magnetic field produced by a field coil carrying a constant current. 3 The Magnetic Field A wire carrying a current generates a magnetic field B whose magnitude and direction at each point in space depend on the length and shape of the wire, the current flowing through the wire, and the location of the point at which the field is determined. A convenient way to depict the pattern of the magnetic field is to draw a lines such that each line is always parallel to the magnetic field B. The pattern of lines shows the direction of the magnetic

2 HB Magnetic Field of a Circular Coil Lab 12 2 field everywhere in space. The intensity of the field is indicated by having the density of the lines show the strength of the field. In certain cases it is easy from the symmetry of the situation to deduce the nature of such a field pattern. For instance, the magnetic field pattern around a long straight current-carrying wire must describe circles centered on the wire, as shown in Fig. 1. The direction of the field is given by the right hand rule. The strength of the field decreases with increasing distance from the wire. This is shown by drawing the lines further apart from each other where the field is weaker. Suppose that the straight wire is bent into the shape of a thin circular coil, with many closely spaced turns of wire. The field at any position is the sum of contributions from the many short elements of wire composing the entire length of wire. You can deduce the general nature of the field pattern of this coil from the magnetic field produced by short length of wire. As shown in Fig. 2, the field contributions of all the short elements add together near the center of the coil to produce a field whose strength is much greater than that of any one element. Furthermore, from the symmetry, you may deduce that the direction of the field is along the axis of the coil. The direction of the axial field is given by the right hand rule where the fingers curl around the coil in the direction of the current and the extended thumb points in the direction of the field. The field in the central region is comparatively strong and uniform. Just outside the coil, very near the wire, the field is due predominantly to the closest portion of the wire, with the current from the far side of the coil contributing relatively little owing to the greater distance of the source. The field has the opposite direction to the field in the center of the coil. Farther out to the side of the coil, the distances from the near portion of the wire and far portion are not so very different, so the fields contributed from each have nearly the same strength but opposite directions. The field from the far side nearly cancels the field from the near side, and the strength of the field must decrease much more rapidly with distance from the coil than if we had a single length of wire. For a long straight wire the field varies as the inverse distance from the wire, and for large distances from a coil it varies as the inverse cube of the distance. 4 Theory Except along the axis, the magnetic field of a circular coil cannot be expressed in closed form. Along the coil axis, if the origin of the coordinates is taken at the center of the coil and if the z axis is taken along the coil axis, the magnitude of the magnetic field B, which points in the z direction, is given by where B is in tesla if B = µ 0Na 2 I, (1) 2(a 2 + z 2 3/2 ) µ 0 =4π 10 7 is the vacuum permeability, N is the number of turns of the field coil, I is the current in the wire, in amperes, a is the radius of the coil in meters, and z is the axial distance in meters from the center of the coil.

3 HB Magnetic Field of a Circular Coil Lab The Experiment Summarizing, the sinusoidally varying current in the field coil produces a magnetic field that varies sinusoidally with time. The part of the magnetic field that threads through the search coil produces a sinusoidally varying voltage in the search coil. This voltage will be measured on an oscilloscope and will be used to determine the magnetic field. The voltage generated in the search coil is due to electromagnetic induction. Assume that the search coil is small and that the magnetic field at a given instant of time is approximately uniform over the area of the search coil. For this situation the flux Φ of the vector B through the search coil is defined as the product of the area A of the coil times the component of B normal to the plane of the search coil. Let the angle between B and the normal to the plane of the search coil be α. See Fig. 3. The flux through the search coil is then Φ = AB cos α. Faraday s law of induction then gives for the voltage induced in one turn of the search coil as dφ.ifn is the number of turns in the search coil, the voltage V induced in the search dt coil is V = N dφ = NAcos αdb dt dt. (2) Due to the current induced in the search coil the search coil produces its own magnetic field. The minus sign in Eq.(2) means that this magnetic field produced by the search coil opposes the change in the magnetic field produced by the field coil. This is an application of Lenz s law. (The voltage in the search coil is produced by a non-conservative electric field whose line integral is called an electromotive force, or emf. The unit of emf is the volt (V), which is work per unit charge.) Eq.(2) uses SI units. The voltage is expressed in volts, the flux in webers, the magnetic field in tesla, the area in square meters, and the time in seconds. The current I through the field coil varies sinusoidally with time so the magnetic field B will also vary sinusoidally with time. The magnitude of the magnetic field B( r, t) produced by the field coil can be written as B = 1 2 B pp cos ωt, (3) where B pp is the peak to peak value of the magnetic field and ω is the angular frequency of the alternating current. B pp = B pp ( r) depends on position but not on the time. Combining Eq.(2) and Eq.(3), the voltage across the search coil becomes V = 1 2 ωnab pp cos α sin ωt. (4) Let V pp be the peak to peak value of the measured voltage for α =0orπ. Then from Eq.(4) V pp = ωnab pp. (5) The voltage V pp will be measured on the oscilloscope. To measure V pp, rotate the search coil so that maximum amplitude of the signal is seen on the scope screen. Use the most sensitive VOLT/CM scope setting that will keep the pattern wholly on the screen and use the grid on the scope screen to determine V pp. Recall that the 2 position knobs on the scope, which move the trace up-down and left-right, are useful in moving the trace to an appropriate place on the scope grid. Be sure the variable gain knob of the scope is fully clockwise in the calibrated

4 HB Magnetic Field of a Circular Coil Lab 12 4 position. To determine the direction of the magnetic field, it is more accurate to rotate the coil so that a zero or minimum signal is seen rather than to look for a maximum. As you minimize the signal on the scope you can increase the gain of the scope. Add or subtract 90 deg to the orientation of the search coil to find the direction of the field. In this experiment the peak to peak current I pp through the field coil is held constant. For a constant I pp the magnetic field B pp is proportional to V pp,sothatwecanwriteb pp = KV pp, where K is a constant. To determine K, measurev pp at a point on the axis of the coil 15 cm from the center and then use Eq.(1) to calculate the field at that point. The field coil current I pp is measured with a Fluke multimeter. 6 Apparatus A schematic drawing of the apparatus is shown in Fig. 4. The search coil is free to slide along a Lucite arm, its distance from the center of the field coil being indicated by a scale along the arm. The angular orientation of the search coil relative to the arm is indicated by a protractor. The Lucite arm is free to turn about a point at the center of the field coil. Another protractor indicates the angular orientation of the Lucite arm. The protractor for the arm reads 90 deg when the arm is along the axis of the field coil. (It would be preferable if the reading was 0 deg.) When you record the data, record what the protractor says and make a note as to what it means. You also have the option of using the top scale or the bottom scale of the protractor. Again, choose one and make a note of your choice. Similar remarks apply to the protractor that measures the angle between the axis of the search coil and the arm. Be sure you know what the angles you write down mean. Is the search coil axis tilted toward the axis or toward the plane of the field coil? 7 Preparation and Calibration 1. Connect the Wavetek oscillator to a series combination of the field coil and the Fluke multimeter. On the multimeter, use the COM and 300 ma receptacles. Turn on the multimeter by setting the dial at Ã, which stands for ac amperes. Set the oscillator s frequency to 2,000 Hz. 2. Connect the search coil to the vertical input of the oscilloscope, connecting the shield wire to ground. (Note the tab with GND on the double banana plug.) 3. Turn on the scope and oscillator. Set the amplitude knob on the oscillator not quite fully clockwise. This will enable you to adjust the current in the field coil both up and down if necessary. Turn the variable gain knob on the scope fully clockwise to the calibrated position and leave it there for the whole experiment. 4. Measure the average diameter (2a) of the field coil and record the number of turns (N) of the field coil. 5. Record the current through the coil by using the Fluke. This is an rms (root mean square) value, I rms. ConvertittoapeaktopeakvalueI pp by multiplying by 2 2. Check the current from time to time, and if you find that it has drifted from the initial value, reset it by using the amplitude knob on the oscillator.

5 HB Magnetic Field of a Circular Coil Lab Calculate the magnetic field B pp of the coil for a point on its axis at a distance of 15 cm from its center for a current of I pp.seeeq.(1). 7. Set the coil at the calibration position (on the coil axis and 15 cm from the center) and measure V pp, the peak to peak search coil voltage, on the scope. A reminder. Rotate the search coil to obtain the maximum signal. Determine the constant K in the equation B pp = KV pp. 8 Response of Search Coil At the calibration position on the axis of the field coil, determine how the voltage V pp varies as a function of the orientation of the search coil relative to the field. Take readings at about every 10 degrees by rotating the search coil without moving it. You should observe the measured voltage varying from a maximum to a minimum (ideally zero) as the search coil s orientation α changes by 90 degrees. Plot you data. Does it follow a cosine curve as suggested by Eq.(2). Is the direction of the magnetic field what you expect? 9 Determining the Field The coordinates and geometry are shown in Fig. 5. The center of the field coil is taken as the origin of polar coordinates (r, θ, φ). The axis of the field coil coincides with the z axis. The vector r, not shown in Fig. 5, is assumed to lie along the Lucite arm with the tip of r at the center of the search coil. The axis of the search coil is shown and is assumed to be along the direction of the magnetic field B. The angle between the magnetic field and the Lucite arm is β. It is clear from the symmetry of the field coil that the magnetic field is axially symmetric and does not depend on the angle φ shown in Fig. 5. You need only record field readings for various values of the radial (r) and angular (θ) coordinates. The angle β should be between -90 deg and +90 deg. When β =0thefieldB is parallel to the Lucite arm. Use the convention that for β positive the field tilts toward the plane of the field coil and for β negative the field tilts toward the axis of the field coil. At a given position of the search coil s center, rotate the search coil to produce the maximum scope signal, V pp. Record V pp, r, and the arm s protractor readings. Calculate B pp and θ. Now rotate the search coil for minimum signal and record the search coil s protractor reading. Calculate β. 1. Make a few measurements of the field at a given r and θ but for different values of φ to convince yourself that the field does not depend on φ. 2. To get a feel for what the field looks like, make a few measurements of B pp scattered over the complete range of r and θ available to you. Estimate the number of measurements you can make in the time available and how these measurements should be distributed in space. It might be convenient to make your measurements at a fixed number of values of the coordinate r, but to make more measurements for the larger r s. When you have a reasonable plan, make your measurements. 3. Plot your data on a piece of paper as shown in Fig. 6 using appropriate distance scales. This rough sketch shows only a very few points. Your Fig. should have more. Make

6 HB Magnetic Field of a Circular Coil Lab 12 6 your plot as large as feasible. Draw a vector of appropriate magnitude and direction at each measuring position. The tails of the arrows should be at the field points. Make a few copies of your Fig. Then using a colored pencil or pen draw field lines on the copies to indicate the general aspects of the field pattern revealed by your measurements. Submit your best drawing with your report. 4. Comment on the accuracy of your measurements. What factor or factors limit the accuracy the most? 5. How rapidly does the field intensity decrease with distance from the center of the coil (a) along the coil s axis and (b) to the side of the coil? 6. Why do you think it is more accurate to measure the direction of the field by looking for a minimum and not a maximum in V pp? 7. How big is the earth s magnetic field, and how does its magnitude compare to the fields produced in this experiment? Why doesn t the earth s field interfere with your measurements? Do you think the technique used in this experiment that separates out the earth s field and the coil s field could be used in other experiments? 10 Comment The voltage and emf induced in the search coil by the field coil is described by a parameter called the mutual inductance (M). M depends on the shape and number of turns of the two coils, their separation, and their relative orientation. If i F is the current in the field coil, the voltage induced in the search coil is M di F dt. If i S is the current in the search coil, the voltage induced in the field coil is M i S dt. In this experiment, due to the large input resistance of the oscilloscope, the current in the search coil is so low that the voltage induced in the field coil by the search coil is negligible. Both the field coil and the search coil have self inductance. A coil s self inductance is due to its own magnetic field threading through the coil. L depends on the shape and number of turns of the coil. If L is the self inductance of the coil and i is the current through the coil, the voltage induced in the coil is L di. In this experiment the voltage induced in the dt search coil by its self inductance is negligible. 11 Finishing Up Please leave the bench as you found it. Thank you.

7

8

9

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law. 260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this

More information

Electrical Resonance

Electrical Resonance Electrical Resonance (R-L-C series circuit) APPARATUS 1. R-L-C Circuit board 2. Signal generator 3. Oscilloscope Tektronix TDS1002 with two sets of leads (see Introduction to the Oscilloscope ) INTRODUCTION

More information

Magnetic Fields and Their Effects

Magnetic Fields and Their Effects Name Date Time to Complete h m Partner Course/ Section / Grade Magnetic Fields and Their Effects This experiment is intended to give you some hands-on experience with the effects of, and in some cases

More information

ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES

ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES The purpose of this lab session is to experimentally investigate the relation between electric field lines of force and equipotential surfaces in two dimensions.

More information

Electromagnetic Induction: Faraday's Law

Electromagnetic Induction: Faraday's Law 1 Electromagnetic Induction: Faraday's Law OBJECTIVE: To understand how changing magnetic fields can produce electric currents. To examine Lenz's Law and the derivative form of Faraday's Law. EQUIPMENT:

More information

Faraday s Law of Induction

Faraday s Law of Induction Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction...10-10.1.1 Magnetic Flux...10-3 10.1. Lenz s Law...10-5 10. Motional EMF...10-7 10.3 Induced Electric Field...10-10 10.4 Generators...10-1

More information

Electromagnetism Laws and Equations

Electromagnetism Laws and Equations Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2

More information

Lesson 3 DIRECT AND ALTERNATING CURRENTS. Task. The skills and knowledge taught in this lesson are common to all missile repairer tasks.

Lesson 3 DIRECT AND ALTERNATING CURRENTS. Task. The skills and knowledge taught in this lesson are common to all missile repairer tasks. Lesson 3 DIRECT AND ALTERNATING CURRENTS Task. The skills and knowledge taught in this lesson are common to all missile repairer tasks. Objectives. When you have completed this lesson, you should be able

More information

Using an Oscilloscope

Using an Oscilloscope Using an Oscilloscope The oscilloscope is used to measure a voltage that changes in time. It has two probes, like a voltmeter. You put these probes on either side of the thing that you want to measure

More information

Direction of Induced Current

Direction of Induced Current Direction of Induced Current Bar magnet moves through coil Current induced in coil A S N v Reverse pole Induced current changes sign B N S v v Coil moves past fixed bar magnet Current induced in coil as

More information

Physics 41, Winter 1998 Lab 1 - The Current Balance. Theory

Physics 41, Winter 1998 Lab 1 - The Current Balance. Theory Physics 41, Winter 1998 Lab 1 - The Current Balance Theory Consider a point at a perpendicular distance d from a long straight wire carrying a current I as shown in figure 1. If the wire is very long compared

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT 4 Understand single-phase alternating current (ac) theory Single phase AC

More information

Scott Hughes 7 April 2005. Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005. Lecture 15: Mutual and Self Inductance.

Scott Hughes 7 April 2005. Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005. Lecture 15: Mutual and Self Inductance. Scott Hughes 7 April 2005 151 Using induction Massachusetts nstitute of Technology Department of Physics 8022 Spring 2005 Lecture 15: Mutual and Self nductance nduction is a fantastic way to create EMF;

More information

E/M Experiment: Electrons in a Magnetic Field.

E/M Experiment: Electrons in a Magnetic Field. E/M Experiment: Electrons in a Magnetic Field. PRE-LAB You will be doing this experiment before we cover the relevant material in class. But there are only two fundamental concepts that you need to understand.

More information

Reading assignment: All students should read the Appendix about using oscilloscopes.

Reading assignment: All students should read the Appendix about using oscilloscopes. 10. A ircuits* Objective: To learn how to analyze current and voltage relationships in alternating current (a.c.) circuits. You will use the method of phasors, or the vector addition of rotating vectors

More information

Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil

Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2006 Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil OBJECTIVES 1. To learn how to visualize magnetic field lines

More information

1. The diagram below represents magnetic lines of force within a region of space.

1. The diagram below represents magnetic lines of force within a region of space. 1. The diagram below represents magnetic lines of force within a region of space. 4. In which diagram below is the magnetic flux density at point P greatest? (1) (3) (2) (4) The magnetic field is strongest

More information

Force on Moving Charges in a Magnetic Field

Force on Moving Charges in a Magnetic Field [ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

More information

A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS

A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS Joseph J. Stupak Jr, Oersted Technology Tualatin, Oregon (reprinted from IMCSD 24th Annual Proceedings 1995) ABSTRACT The

More information

Experiment 8: Undriven & Driven RLC Circuits

Experiment 8: Undriven & Driven RLC Circuits Experiment 8: Undriven & Driven RLC Circuits Answer these questions on a separate sheet of paper and turn them in before the lab 1. RLC Circuits Consider the circuit at left, consisting of an AC function

More information

Physics 221 Experiment 5: Magnetic Fields

Physics 221 Experiment 5: Magnetic Fields Physics 221 Experiment 5: Magnetic Fields August 25, 2007 ntroduction This experiment will examine the properties of magnetic fields. Magnetic fields can be created in a variety of ways, and are also found

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

More information

Ampere's Law. Introduction. times the current enclosed in that loop: Ampere's Law states that the line integral of B and dl over a closed path is 0

Ampere's Law. Introduction. times the current enclosed in that loop: Ampere's Law states that the line integral of B and dl over a closed path is 0 1 Ampere's Law Purpose: To investigate Ampere's Law by measuring how magnetic field varies over a closed path; to examine how magnetic field depends upon current. Apparatus: Solenoid and path integral

More information

ElectroMagnetic Induction. AP Physics B

ElectroMagnetic Induction. AP Physics B ElectroMagnetic Induction AP Physics B What is E/M Induction? Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a complete circuit, a current. Michael Faraday

More information

Chapter 27 Magnetic Field and Magnetic Forces

Chapter 27 Magnetic Field and Magnetic Forces Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged

More information

Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6. Instructions: 1. In the formula F = qvxb:

Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6. Instructions: 1. In the formula F = qvxb: Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6 Signature Name (Print): 4 Digit ID: Section: Instructions: Answer all questions 24 multiple choice questions. You may need to do some calculation.

More information

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H). INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes

More information

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 30 - Magnetic Fields and Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Objectives: After completing this module, you should

More information

Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance. Your name Lab section

Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance. Your name Lab section Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance Your name Lab section 1. What do you investigate in this lab? 2. Two straight wires are in parallel and carry electric currents in opposite directions

More information

The DC Motor. Physics 1051 Laboratory #5 The DC Motor

The DC Motor. Physics 1051 Laboratory #5 The DC Motor The DC Motor Physics 1051 Laboratory #5 The DC Motor Contents Part I: Objective Part II: Introduction Magnetic Force Right Hand Rule Force on a Loop Magnetic Dipole Moment Torque Part II: Predictions Force

More information

ELECTRON SPIN RESONANCE Last Revised: July 2007

ELECTRON SPIN RESONANCE Last Revised: July 2007 QUESTION TO BE INVESTIGATED ELECTRON SPIN RESONANCE Last Revised: July 2007 How can we measure the Landé g factor for the free electron in DPPH as predicted by quantum mechanics? INTRODUCTION Electron

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

5. Measurement of a magnetic field

5. Measurement of a magnetic field H 5. Measurement of a magnetic field 5.1 Introduction Magnetic fields play an important role in physics and engineering. In this experiment, three different methods are examined for the measurement of

More information

Experiment 7: Forces and Torques on Magnetic Dipoles

Experiment 7: Forces and Torques on Magnetic Dipoles MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics 8. Spring 5 OBJECTIVES Experiment 7: Forces and Torques on Magnetic Dipoles 1. To measure the magnetic fields due to a pair of current-carrying

More information

Force on a square loop of current in a uniform B-field.

Force on a square loop of current in a uniform B-field. Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis

More information

Lab E1: Introduction to Circuits

Lab E1: Introduction to Circuits E1.1 Lab E1: Introduction to Circuits The purpose of the this lab is to introduce you to some basic instrumentation used in electrical circuits. You will learn to use a DC power supply, a digital multimeter

More information

Alternating-Current Circuits

Alternating-Current Circuits hapter 1 Alternating-urrent ircuits 1.1 A Sources... 1-1. Simple A circuits... 1-3 1..1 Purely esistive load... 1-3 1.. Purely Inductive oad... 1-5 1..3 Purely apacitive oad... 1-7 1.3 The Series ircuit...

More information

Physics 25 Exam 3 November 3, 2009

Physics 25 Exam 3 November 3, 2009 1. A long, straight wire carries a current I. If the magnetic field at a distance d from the wire has magnitude B, what would be the the magnitude of the magnetic field at a distance d/3 from the wire,

More information

Magnetism Basics. Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment. Net Effect = Zero!

Magnetism Basics. Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment. Net Effect = Zero! Magnetism Basics Source: electric currents Magnetic Domains: atomic regions of aligned magnetic poles Random Alignment Ferromagnetic Alignment Net Effect = Zero! Net Effect = Additive! Bipolar: all magnets

More information

Chapter 12 Driven RLC Circuits

Chapter 12 Driven RLC Circuits hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

More information

104 Practice Exam 2-3/21/02

104 Practice Exam 2-3/21/02 104 Practice Exam 2-3/21/02 1. Two electrons are located in a region of space where the magnetic field is zero. Electron A is at rest; and electron B is moving westward with a constant velocity. A non-zero

More information

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be

More information

Magnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise.

Magnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise. Magnetism 1. An electron which moves with a speed of 3.0 10 4 m/s parallel to a uniform magnetic field of 0.40 T experiences a force of what magnitude? (e = 1.6 10 19 C) a. 4.8 10 14 N c. 2.2 10 24 N b.

More information

Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil

Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2009 Experiment 3: Magnetic Fields of a Bar Magnet and Helmholtz Coil OBJECTIVES 1. To learn how to visualize magnetic field lines

More information

AP2 Magnetism. (c) Explain why the magnetic field does no work on the particle as it moves in its circular path.

AP2 Magnetism. (c) Explain why the magnetic field does no work on the particle as it moves in its circular path. A charged particle is projected from point P with velocity v at a right angle to a uniform magnetic field directed out of the plane of the page as shown. The particle moves along a circle of radius R.

More information

EE 1202 Experiment #4 Capacitors, Inductors, and Transient Circuits

EE 1202 Experiment #4 Capacitors, Inductors, and Transient Circuits EE 1202 Experiment #4 Capacitors, Inductors, and Transient Circuits 1. Introduction and Goal: Exploring transient behavior due to inductors and capacitors in DC circuits; gaining experience with lab instruments.

More information

Chapter 22 Magnetism

Chapter 22 Magnetism 22.6 Electric Current, Magnetic Fields, and Ampere s Law Chapter 22 Magnetism 22.1 The Magnetic Field 22.2 The Magnetic Force on Moving Charges 22.3 The Motion of Charged particles in a Magnetic Field

More information

45. The peak value of an alternating current in a 1500-W device is 5.4 A. What is the rms voltage across?

45. The peak value of an alternating current in a 1500-W device is 5.4 A. What is the rms voltage across? PHYS Practice Problems hapters 8- hapter 8. 45. The peak value of an alternating current in a 5-W device is 5.4 A. What is the rms voltage across? The power and current can be used to find the peak voltage,

More information

Chapter 11. Inductors ISU EE. C.Y. Lee

Chapter 11. Inductors ISU EE. C.Y. Lee Chapter 11 Inductors Objectives Describe the basic structure and characteristics of an inductor Discuss various types of inductors Analyze series inductors Analyze parallel inductors Analyze inductive

More information

6/2016 E&M forces-1/8 ELECTRIC AND MAGNETIC FORCES. PURPOSE: To study the deflection of a beam of electrons by electric and magnetic fields.

6/2016 E&M forces-1/8 ELECTRIC AND MAGNETIC FORCES. PURPOSE: To study the deflection of a beam of electrons by electric and magnetic fields. 6/016 E&M forces-1/8 ELECTRIC AND MAGNETIC FORCES PURPOSE: To study the deflection of a beam of electrons by electric and magnetic fields. APPARATUS: Electron beam tube, stand with coils, power supply,

More information

ε: Voltage output of Signal Generator (also called the Source voltage or Applied

ε: Voltage output of Signal Generator (also called the Source voltage or Applied Experiment #10: LR & RC Circuits Frequency Response EQUIPMENT NEEDED Science Workshop Interface Power Amplifier (2) Voltage Sensor graph paper (optional) (3) Patch Cords Decade resistor, capacitor, and

More information

Inductors & Inductance. Electronic Components

Inductors & Inductance. Electronic Components Electronic Components Induction In 1824, Oersted discovered that current passing though a coil created a magnetic field capable of shifting a compass needle. Seven years later, Faraday and Henry discovered

More information

F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below.

F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below. PHYSICS 176 UNIVERSITY PHYSICS LAB II Experiment 9 Magnetic Force on a Current Carrying Wire Equipment: Supplies: Unit. Electronic balance, Power supply, Ammeter, Lab stand Current Loop PC Boards, Magnet

More information

SERIES-PARALLEL DC CIRCUITS

SERIES-PARALLEL DC CIRCUITS Name: Date: Course and Section: Instructor: EXPERIMENT 1 SERIES-PARALLEL DC CIRCUITS OBJECTIVES 1. Test the theoretical analysis of series-parallel networks through direct measurements. 2. Improve skills

More information

How To Understand Electron Spin Resonance

How To Understand Electron Spin Resonance HB 10-24-08 Electron Spin Resonance Lab 1 Electron Spin Resonance Equipment Electron Spin Resonance apparatus, leads, BK oscilloscope, 15 cm ruler for setting coil separation Reading Review the Oscilloscope

More information

PHYS 222 Spring 2012 Final Exam. Closed books, notes, etc. No electronic device except a calculator.

PHYS 222 Spring 2012 Final Exam. Closed books, notes, etc. No electronic device except a calculator. PHYS 222 Spring 2012 Final Exam Closed books, notes, etc. No electronic device except a calculator. NAME: (all questions with equal weight) 1. If the distance between two point charges is tripled, the

More information

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name Name If you think that no correct answer is provided, give your answer, state your reasoning briefly; append additional sheet of paper if necessary. 1. A particle (q = 5.0 nc, m = 3.0 µg) moves in a region

More information

Candidate Number. General Certificate of Education Advanced Level Examination June 2014

Candidate Number. General Certificate of Education Advanced Level Examination June 2014 entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 214 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Wednesday

More information

Slide 1 / 26. Inductance. 2011 by Bryan Pflueger

Slide 1 / 26. Inductance. 2011 by Bryan Pflueger Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

More information

Chapter 22: Electric motors and electromagnetic induction

Chapter 22: Electric motors and electromagnetic induction Chapter 22: Electric motors and electromagnetic induction The motor effect movement from electricity When a current is passed through a wire placed in a magnetic field a force is produced which acts on

More information

Understanding Poles and Zeros

Understanding Poles and Zeros MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

More information

DC motors: dynamic model and control techniques

DC motors: dynamic model and control techniques DC motors: dynamic model and control techniques Luca Zaccarian Contents 1 Magnetic considerations on rotating coils 1 1.1 Magnetic field and conductors.......................... 1 1.2 The magneto-motive

More information

Review Questions PHYS 2426 Exam 2

Review Questions PHYS 2426 Exam 2 Review Questions PHYS 2426 Exam 2 1. If 4.7 x 10 16 electrons pass a particular point in a wire every second, what is the current in the wire? A) 4.7 ma B) 7.5 A C) 2.9 A D) 7.5 ma E) 0.29 A Ans: D 2.

More information

Objectives. Capacitors 262 CHAPTER 5 ENERGY

Objectives. Capacitors 262 CHAPTER 5 ENERGY Objectives Describe a capacitor. Explain how a capacitor stores energy. Define capacitance. Calculate the electrical energy stored in a capacitor. Describe an inductor. Explain how an inductor stores energy.

More information

NUCLEAR MAGNETIC RESONANCE. Advanced Laboratory, Physics 407, University of Wisconsin Madison, Wisconsin 53706

NUCLEAR MAGNETIC RESONANCE. Advanced Laboratory, Physics 407, University of Wisconsin Madison, Wisconsin 53706 (revised 4/21/03) NUCLEAR MAGNETIC RESONANCE Advanced Laboratory, Physics 407, University of Wisconsin Madison, Wisconsin 53706 Abstract This experiment studies the Nuclear Magnetic Resonance of protons

More information

Linear DC Motors. 15.1 Magnetic Flux. 15.1.1 Permanent Bar Magnets

Linear DC Motors. 15.1 Magnetic Flux. 15.1.1 Permanent Bar Magnets Linear DC Motors The purpose of this supplement is to present the basic material needed to understand the operation of simple DC motors. This is intended to be used as the reference material for the linear

More information

AC generator theory. Resources and methods for learning about these subjects (list a few here, in preparation for your research):

AC generator theory. Resources and methods for learning about these subjects (list a few here, in preparation for your research): AC generator theory This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

More information

INTERFERENCE OF SOUND WAVES

INTERFERENCE OF SOUND WAVES 1/2016 Sound 1/8 INTERFERENCE OF SOUND WAVES PURPOSE: To measure the wavelength, frequency, and propagation speed of ultrasonic sound waves and to observe interference phenomena with ultrasonic sound waves.

More information

INTERFERENCE OF SOUND WAVES

INTERFERENCE OF SOUND WAVES 2011 Interference - 1 INTERFERENCE OF SOUND WAVES The objectives of this experiment are: To measure the wavelength, frequency, and propagation speed of ultrasonic sound waves. To observe interference phenomena

More information

Module P4.4 Electromagnetic induction

Module P4.4 Electromagnetic induction F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P4.4 Electromagnetic induction 1 Opening items 1.1 Module introduction 1.2 Fast track questions 1.3 Ready to study? 2 Introducing

More information

Magnetic electro-mechanical machines

Magnetic electro-mechanical machines Magnetic electro-mechanical machines Lorentz Force A magnetic field exerts force on a moving charge. The Lorentz equation: f = q(e + v B) f: force exerted on charge q E: electric field strength v: velocity

More information

FORCE ON A CURRENT IN A MAGNETIC FIELD

FORCE ON A CURRENT IN A MAGNETIC FIELD 7/16 Force current 1/8 FORCE ON A CURRENT IN A MAGNETIC FIELD PURPOSE: To study the force exerted on an electric current by a magnetic field. BACKGROUND: When an electric charge moves with a velocity v

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has

More information

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering 2. A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of

More information

Solution Derivations for Capa #11

Solution Derivations for Capa #11 Solution Derivations for Capa #11 Caution: The symbol E is used interchangeably for energy and EMF. 1) DATA: V b = 5.0 V, = 155 Ω, L = 8.400 10 2 H. In the diagram above, what is the voltage across the

More information

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad. Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

More information

Measuring Impedance and Frequency Response of Guitar Pickups

Measuring Impedance and Frequency Response of Guitar Pickups Measuring Impedance and Frequency Response of Guitar Pickups Peter D. Hiscocks Syscomp Electronic Design Limited phiscock@ee.ryerson.ca www.syscompdesign.com April 30, 2011 Introduction The CircuitGear

More information

12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance?

12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance? 12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its self-inductance? From Equation 32-5, L = -E=(dI =dt) = 40 V=(100 A/s) = 0.4 H. 15. A cardboard

More information

Induced voltages and Inductance Faraday s Law

Induced voltages and Inductance Faraday s Law Induced voltages and Inductance Faraday s Law concept #1, 4, 5, 8, 13 Problem # 1, 3, 4, 5, 6, 9, 10, 13, 15, 24, 23, 25, 31, 32a, 34, 37, 41, 43, 51, 61 Last chapter we saw that a current produces a magnetic

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

More information

Electric Field Mapping Lab 3. Precautions

Electric Field Mapping Lab 3. Precautions HB 09-25-07 Electric Field Mapping Lab 3 1 Electric Field Mapping Lab 3 Equipment mapping board, U-probe, resistive boards, templates, dc voltmeter (431B), 4 long leads, 16 V dc for wall strip Reading

More information

Reflection and Refraction

Reflection and Refraction Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

More information

MAG Magnetic Fields revised July 24, 2012

MAG Magnetic Fields revised July 24, 2012 MAG Magnetic Fields revised July 24, 2012 (You will do two experiments; this one (in Rock 402) and the Magnetic Induction experiment (in Rock 403). Sections will switch rooms and experiments half-way through

More information

Lab Exercise 1: Acoustic Waves

Lab Exercise 1: Acoustic Waves Lab Exercise 1: Acoustic Waves Contents 1-1 PRE-LAB ASSIGNMENT................. 2 1-3.1 Spreading Factor: Spherical Waves........ 2 1-3.2 Interference In 3-D................. 3 1-4 EQUIPMENT........................

More information

Lab 4: Magnetic Force on Electrons

Lab 4: Magnetic Force on Electrons Lab 4: Magnetic Force on Electrons Introduction: Forces on particles are not limited to gravity and electricity. Magnetic forces also exist. This magnetic force is known as the Lorentz force and it is

More information

Magnetic Dipoles. Recall that an electric dipole consists of two equal but opposite charges separated by some distance, such as in

Magnetic Dipoles. Recall that an electric dipole consists of two equal but opposite charges separated by some distance, such as in MAGNETISM History of Magnetism Bar Magnets Magnetic Dipoles Magnetic Fields Magnetic Forces on Moving Charges and Wires Electric Motors Current Loops and Electromagnets Solenoids Sources of Magnetism Spin

More information

Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

More information

Chapter 19: Magnetic Forces and Fields

Chapter 19: Magnetic Forces and Fields Chapter 19: Magnetic Forces and Fields Magnetic Fields Magnetic Force on a Point Charge Motion of a Charged Particle in a Magnetic Field Crossed E and B fields Magnetic Forces on Current Carrying Wires

More information

Chapter 21. Magnetic Forces and Magnetic Fields

Chapter 21. Magnetic Forces and Magnetic Fields Chapter 21 Magnetic Forces and Magnetic Fields 21.1 Magnetic Fields The needle of a compass is permanent magnet that has a north magnetic pole (N) at one end and a south magnetic pole (S) at the other.

More information

Physics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5

Physics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5 Solutions to Homework Questions 5 Chapt19, Problem-2: (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat

More information

Fraunhofer Diffraction

Fraunhofer Diffraction Physics 334 Spring 1 Purpose Fraunhofer Diffraction The experiment will test the theory of Fraunhofer diffraction at a single slit by comparing a careful measurement of the angular dependence of intensity

More information

E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051 NUCLEAR MAGNETIC RESONANCE

E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051 NUCLEAR MAGNETIC RESONANCE E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051 NUCLEAR MAGNETIC RESONANCE References for Nuclear Magnetic Resonance 1. Slichter, Principles of Magnetic Resonance, Harper and Row, 1963. chapter

More information

BASIC ELECTRONICS AC CIRCUIT ANALYSIS. December 2011

BASIC ELECTRONICS AC CIRCUIT ANALYSIS. December 2011 AM 5-202 BASIC ELECTRONICS AC CIRCUIT ANALYSIS December 2011 DISTRIBUTION RESTRICTION: Approved for Pubic Release. Distribution is unlimited. DEPARTMENT OF THE ARMY MILITARY AUXILIARY RADIO SYSTEM FORT

More information

Application Note. So You Need to Measure Some Inductors?

Application Note. So You Need to Measure Some Inductors? So You Need to Measure Some nductors? Take a look at the 1910 nductance Analyzer. Although specifically designed for production testing of inductors and coils, in addition to measuring inductance (L),

More information

Lab 3 - DC Circuits and Ohm s Law

Lab 3 - DC Circuits and Ohm s Law Lab 3 DC Circuits and Ohm s Law L3-1 Name Date Partners Lab 3 - DC Circuits and Ohm s Law OBJECTIES To learn to apply the concept of potential difference (voltage) to explain the action of a battery in

More information

Magnetic Fields and Forces. AP Physics B

Magnetic Fields and Forces. AP Physics B Magnetic ields and orces AP Physics acts about Magnetism Magnets have 2 poles (north and south) Like poles repel Unlike poles attract Magnets create a MAGNETIC IELD around them Magnetic ield A bar magnet

More information

Experiment 5: Magnetic Fields of a Bar Magnet and of the Earth

Experiment 5: Magnetic Fields of a Bar Magnet and of the Earth MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2005 Experiment 5: Magnetic Fields of a Bar Magnet and of the Earth OBJECTIVES 1. To examine the magnetic field associated with a

More information

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

More information

DIRECT CURRENT GENERATORS

DIRECT CURRENT GENERATORS DIRECT CURRENT GENERATORS Revision 12:50 14 Nov 05 INTRODUCTION A generator is a machine that converts mechanical energy into electrical energy by using the principle of magnetic induction. This principle

More information

Interferometers. OBJECTIVES To examine the operation of several kinds of interferometers. d sin = n (1)

Interferometers. OBJECTIVES To examine the operation of several kinds of interferometers. d sin = n (1) Interferometers The true worth of an experimenter consists in his pursuing not only what he seeks in his experiment, but also what he did not seek. Claude Bernard (1813-1878) OBJECTIVES To examine the

More information